Adaptive test statistics for ergodic diffusion processes sampled at discrete times

We consider adaptive test statistics for ergodic diffusion processes based on discrete observations. Since an exact likelihood function for the discretely observed diffusion process cannot been generally obtained, the quasi-likelihood function based on the Ito–Taylor expansion is used and three kinds of test statistics, likelihood ratio type test statistic, Wald type test statistic and Rao׳s score type test statistic, for diffusion processes are proposed. It is shown that the test statistics converge in distribution to χ2 (the chi-squared distribution) under null hypothesis and the tests are consistent. Moreover, we prove that the test statistics converge in distribution to the noncentral χ2 under local alternative hypothesis.